If $$x - \frac{1}{x} = y, y - \frac{1}{y} = z, z - \frac{1}{z} = x$$, there which of the following equations are true?
A. $$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0$$
B. $$\frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{z^{2}} = 8$$
C. $$\frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx} = -3$$
Choose the correct answer from the options given below:
CMAT Linear Equations Questions
It is given,
$$x-\frac{1}{x}=y$$, $$y-\frac{1}{y}=z$$ and $$z-\frac{1}{z}=x$$
Adding all the given equations, we get
$$x\ +\ y\ +\ z\ -\ \frac{1}{x}-\frac{1}{y}-\frac{1}{z}=y+z+x$$
$$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$$
Therefore, equation A is correct.
$$x-\frac{1}{x}=y$$, $$y-\frac{1}{y}=z$$ and $$z-\frac{1}{z}=x$$
Squaring on both the sides and adding three resultant equations, we get
$$x^2+\frac{1}{x^2}-2+y^2+\frac{1}{y^2}-2+z^2+\frac{1}{z^2}-2=y^2+z^2+x^2$$
$$\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=6$$
Therefore, equation B is incorrect.
$$(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})^2=0$$
On expanding, $$(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+\frac{2}{xy}+\frac{2}{yz}+\frac{2}{xz})=0$$
We know that $$\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=6$$
So, $$2(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}) = -6$$
$$(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}) = -3$$
Therefore, equation C is correct.
The answer is option B.
Frequently Asked Questions
Yes, Linear Equations is an important topic in the Quantitative Aptitude section of CMAT. It tests a candidate's algebraic concepts, logical reasoning, and ability to solve equation-based problems accurately.
The number of Linear Equations questions varies from year to year. CMAT does not prescribe a fixed number of questions from any specific Quantitative Aptitude topic.
CMAT may include questions on linear equations in one variable, simultaneous linear equations, word problems, substitution and elimination methods, and algebraic applications.
Learn the fundamental concepts of linear equations, practice different solving techniques regularly, strengthen algebraic manipulation skills, and solve previous year questions and mock tests to improve speed and accuracy.
Most Linear Equations questions in CMAT are of easy to moderate difficulty. With conceptual clarity and regular practice, candidates can solve them efficiently and accurately.
Cracku's CMAT Linear Equations Questions provide topic-wise practice, detailed solutions, and exam-oriented problems that help candidates strengthen algebraic concepts, improve accuracy, and perform better in CMAT 2027.